study of torsional vibrations of composite poroelastic

© 2020 IAU, Arak Branch. All rights reserved.

Journal of Solid Mechanics Vol. 12, No. 3 (2020) pp. 649-662

DOI: 10.22034/jsm.2020.1885789.1529

Study of Torsional Vibrations of Composite Poroelastic Spherical Shell-Biot’s Extension Theory

R. Gurijala*, M. Reddy Perati

Department of Mathematics, Kakatiya University, Warangal, Telangana, India

Received 28 June 2020; accepted 24 August 2020

ABSTRACT

Torsional vibrations of composite poroelastic dissipative spherical

shell are investigated in the framework of Biot’s extension theory.

Here composite poroelastic spherical shell consists of two spherical

shells, one is placed on other, and both are made of different

poroelastic materials. Consideration of the stress-free boundaries of

outer surface and the perfect bonding between two shells leads to

complex valued frequency equation. Limiting case when the ratio

of thickness to inner radius is very small is investigated

numerically. In this case, thick walled composite spherical shell

reduces to thin composite spherical shell. For illustration purpose,

four composite materials, namely, Berea sandstone saturated with

water and kerosene, Shale rock saturated with water and kerosene

are employed. The particular cases of a poroelastic solid spherical

shell and poroelastic thick walled hollow spherical shell are

discussed. If the shear viscosity of fluid is neglected, then the

problem reduces to that of classical Biot’s theory. Phase velocity

and attenuation are computed and the results are presented

graphically. Comparison is made between the results of Biot’s

extension theory and that of classical Biot’s theory. It is conclude

that shear viscosity of fluid is causing the discrepancy of the

numerical results.

© 2020 IAU, Arak Branch. All rights reserved.

Keywords: Torsional vibrations; Composite spherical shell;

Frequency equation; Phase velocity; Attenuation.

1 INTRODUCTION

HE analysis of torsional vibrations in poroelastic solids is important in several fields. In the Electrical

Engineering, power needs to be transmitted using a rotating shaft or a coupling, which causes torsional

vibrations. In the case of Automobile Engineering, the auto motives, truck and bus drivelines, recreation vehicles,

and marine drivelines experience torsional vibrations. These vibrations are influenced by material properties and

operating conditions. Thus, torsional vibrations find its vast applications in different branches of Engineering. On

one hand, composite structures are combination of both load bearing as well as framed structures, and most useful

materials. Because of their adaptability to different situations and the relative ease of combination with other

______ *Corresponding author. Tel.:+99 59875482.

E-mail address: [email protected] (R. Gurijala).

T

650 R. Gurijala and M. Reddy Perati

© 2020 IAU, Arak Branch

materials, they serve specific purposes, and exhibit desirable properties. Hence, further investigation of composite

materials is warranted. On other hand, spherical shell structures play vital role in various domains, particularly, in

modern Structural Engineering. In this direction, the following papers are available in the literature. Frequency

equations and mode shapes are presented in analytic form for solid prolate spheroids and thick prolate spheroidal

shells [1]. Heyliger and Pan [2] presented a discrete-layer model and applied to the free vibration of layered

anisotropic spheres with coupling among the elastic, electric, and magnetic fields. Employing Biot’s theory [3],

Shah and Tajuddin [4] discussed torsional vibrations of poroelastic spheroidal shells. In the paper [4], they derived

frequency equations for poroelastic thin spherical shell, thick spherical shell, poroelastic solid sphere, and concluded

that the wave frequency is same in all the three cases. For radial and rotatory vibrations, frequency equations of a

poroelastic composite hollow sphere and a poroelastic composite hollow sphere with rigid core are obtained [5].

Vibrations in poroelastic elliptic cone against the angle made by the major axis of the cone in the spheroconal

coordinate system are studied by Rajitha and Reddy [6, 7]. A comparative study is made between the modes of

composite spherical shell and its ring modes [8]. In the paper [9], authors concluded that the torsional waves are non

dispersive in thin coated hollow poroelastic sphere. In all the above studies, solid saturated with the non-viscous

fluid is considered which may not be realistic in all the solids. In view of this limitation, Sahay [10] has developed

constitutive relations for the case of viscous fluid. In the paper [10], the volume-average equations of motion for the

angular displacement fields are derived. This approach introduces the missing fluid-strain rate term in the Biot’s

extended constitutive relations. Employing Biot’s extension theory of poroelasticity, Solorza and Sahay [11]

investigated the extensional wave in a poroelastic cylinder, wherein the axial motion is compressional in nature, that

is, the direction of propagation is along the axial direction, and the radial motion is shear in nature, as it is being

executed perpendicular to the direction of propagation. Reddy and Rajitha [12, 13] discussed the torsional and radial

vibrations of thick walled hollow poroelastic cylinder in the framework of Biot’s extension theory. To the best of

authors’ knowledge, torsional vibrations in composite poroelastic spherical shell in the framework of Biot’s

extension theory are not yet investigated. Hence in the present work, the same is taken up. Limiting case, when the

ratio between thickness and inner radius is very small is investigated numerically.

In this case, the asymptotic expansions of Bessel functions are employed so that complex valued frequency

equation can be separated into two real parts, which in turn give phase velocity and attenuation. In the particular

case of thick walled hollow spherical shell, phase velocity and attenuation computed as a function of ratio of

thickness to inner radius. In the case of classical Biot’s theory, phase velocity is computed as a function of

frequency. The rest of the paper is organized as follows. In section 2, formulation and solution of the problem are

given. In section 3, the boundary conditions and frequency equation are discussed. Particular cases are discussed in

section 4. Numerical results are presented graphically in section 5. Finally, the conclusion is given in section 6.

2 FORMULATION AND SOLUTION OF THE PROBLEM

Consider a composite poroelastic spherical shell with outer and inner radii 2r and 1r respectively, in the spherical

coordinate system ( , , )r . The shell is made up of two different materials where in inner one is spherical shell of

radius 1( )r and the outer one is spherical shell having uniform thickness 2 1( 0)h r r . For torsional vibrations, the

volume-averaged equation of motion in terms of angular displacement field u expressed as follows [11, 12]:

2 2

02 2

2.i

u uC N u I I

t r r tr t

(1)

In Eq. (1), 21

,f

c

f f f

mC

d d m

0 2

0

( ) 1,

( )

f

c

s s f

mN

d d m

1f

f

dS m

, and 0 s

s

i

d

, 0

is

unperturbed volume fraction of solid, S is tortuosity factor, 2 0

c

m

is Gassmann S -wave velocity,

0 is dry solid

frame shear modulus, is Biot shear coefficient, is saturated frame shear relaxation frequency, i fd is

Study of Torsional Vibrations of Composite Poroelastic…. 651


Biot relaxation frequency, 0 f

bK

is the Biot critical frequency,

0 is porosity, K is permeability, and

f

f

f

is the shear viscosity,

f is the fluid shear viscosity. The matrix 0I is 2 2 matrix whose element (2 2) is

unity and rest of the elements are equal to zero, and I is the second order identity matrix. The vector

u 1 .T T

m i s f

j j j ju u m u u The notation m

ju is the sum of mass-weighted angular displacement of the solid

and fluid phases. The notation

i

ju is the difference of the angular displacements of two phases. The transformation

matrix is defined by1

1

f

s

mm

m

,

fm , sm are the solid and fluid mass fractions, respectively, and ,m r are the

total and reduced densities of the porous medium, respectively, given by

0 0 ,m s f 0 0

1 1 1

r s f ,

12i r is the modified reduced density, and ,s f are the solid and

fluid densities. 12 is the induced mass coefficient which is linked to the tortuosity, as

12 0( 1) .fS In the

frequency domain, Eq. (1) becomes

2 2

2

20,β I u

r

(2)

where

i1( )mm mi

im iiβ C N

, is a non-symmetric second order matrix associated with S motion,

respectively, whose elements are dimensionally equal to velocity squared. The expressions of these parameters are

given in [11, 12]. The notation 0

iI i I

is a 2 2 diagonal matrix associated with the Biot relaxation

frequency

.i For torsional vibrations, the displacement components which are functions of r and t are introduced

as follows:

( ) , 1,2.y yu R F r y β (3)

here [ , ]I IIR r r and [ , ]I IIL I I are the right- and left- hand Eigen vector matrices of the β matrix,

respectively, their explicit expressions are given in [11,12]. These Eigen vectors are orthonormal to each other,

therefore TL R I .

Here1 2

TF( r ) [ F ( r ),F ( r )] , and k is the wave number. The subscript 1y refers to the

inner shell, whereas 2y refers to the outer shell. Substituting Eq. (3) in Eq. (2), and then multiplying TL on both

sides, the equations for S motion are obtained as:

2 2

2

20 1 2yΛ I F( r ) , y , ,

r

(4)

here

2

2

01 2

0

y IT

y II

Λ L R , y , .

β

In the region below Biot relaxation frequency i y I y II( ) , ,

are the fast and slow S wave velocities given by [11]

2 2 1 1 2y I y c y f y f

y i

i d m , y , ,

2 1 1 1 2y II y f y f y f y f

y i y i

i ( d m ) d , y , .

(5)



The solutions are

2 1 1 0 2 2 0 2F ( r ) C J ( pr ) C Y ( pr ) , 1 1 3 0 1F ( r ) C J ( pr ),

2 2 4 0 2 5 0 2F ( r ) C J ( qr ) C Y ( qr ) , 1 2 6 0 1F ( r ) C J ( qr ). (6)

In the above, 1 2 3 4 5C ,C ,C ,C ,C and

6C are constants, 0J and

0Y are the Bessel functions of the first and second

kind of order zero, respectively, and

1 1

2 22 22 2

2 21 2y y

y I y II

p k , q k , y , .

The modified constitutive relations are expressed as [11]

2 1 2

m m

y r y r

y b y b ti i

y r y r

σ uμ , y , ,

σ u

(7)

here0

0 0

0 01 01 2

0 0y y b y s y b

y y

μ , , y , .( )

Eq. (7) can also be expressed as:

2 1 2y r y y y y rσ Ω β u , y , , (8)

here 1 1

1 22

r yu u , y , .r r

y

3 BOUNDARY CONDITIONS AND FREQUENCY EQUATION

Consider the case when outer surface 2r r is stress free and perfect bonding prevails at the interface 1r r . The

boundary conditions in this case are mathematically expressed as follows:

2 2 0m i

r r at 2r r ,

2 1 2 10 0m m i i

r r r r, at 1r r ,

2 1 2 1

m m i i

r r r ru u , u u at 1r r .

(9)

These boundary conditions lead to the following system of homogeneous equations:

lm lA C , l ,m , , , , , . 0 1 2 3 4 5 6 (10)

Eq. (10) is complex valued and is transcendental in nature. Hence, the limiting case h

r

1

1 is considered. In this

case, composite thick walled hollow spherical shell reduces to composite thin spherical shell so that the following

asymptotic approximations for Bessel’s function [13, 14] can be used:

n

n nJ ( x ) cos x sin x ,

x x

2 1 15 1

2 4 8 2 4



n

n nY ( x ) sin x cos x .

x x

2 1 15 1

2 4 8 2 4

In this case, the frequency equation is resolved as:

l m l mc i d ( l ,m , , , , , ). 0 1 2 3 4 5 6 (11)

In Eq. (11), the elements l mc and l md are real valued and are given in the Appendix A.

4 PARTICULAR CASES

The composite poroelastic spherical shell will be reduced to the poroelastic solid spherical shell and the poroelastic

thick walled hollow spherical shell as follows.

4.1 Poroelastic solid spherical shell

When the poroelastic parameters of both outer and inner shells are of same material, then the composite poroelastic

spherical shell results in a solid spherical shell of one material. Then the frequency Eq. (11) reduces to

l m l me i f ( l ,m , ). 0 1 2 (12)

4.2 Poroelastic thick walled hollow spherical shell

Consider the case where the inner shell material parameters are vanishes, the composite poroelastic spherical shell

results in a thick walled hollow poroelastic spherical shell with thickness h ( r r ) 2 1. Then the frequency Eq. (11)

reduces to

l m l mg i h ( l ,m , , , ). 0 1 2 3 4 (13)

The expressions for Eq. (12) and (13) are similar expressions as Appendix A without left subscript.

5 NUMERICAL RESULTS

The real part of Eq. (11) or Eq. (13) gives the phase velocity as in the case of non dissipative solid for composite and

thick walled hollow spherical shell. The attenuation ( Q )1 is computed by using the following equation [11].

( of Im aginary part in Eq.( ) or Eq.( ))Q .

of real part in Eq.( ) or Eq.( )

1 2 11 13

11 13

For the illustration purpose, four types of poroelastic solids namely, composite spherical shell-I, composite

spherical shell-II, composite spherical shell-III and composite spherical shell-IV are employed. In composite

spherical shell-1, outer shell is made up of Berea sandstone saturated with water [13] and inner shell is made up of

Berea sandstone saturated with kerosene [13]. In composite spherical shell- II, the roles of materials are reversed. In

composite spherical shell-III, outer shell is made up of Shale rock saturated with water [15] and inner shell is made

up of Shale rock saturated with kerosene [15]. In composite spherical shell-IV, the roles of materials are reversed.

The physical parameter values are given in Table 1 and Table 2.



Table 1

Parameter values of solids

Table 2

Parameter values of fluids.

Employing these values in Eq. (11), phase velocity and attenuation are computed as a function of frequency, and

the results are depicted in Figs. 1 and 2. Fig. 1 depicts variation of phase velocity with frequency for composite

spherical shell-I, shell-II, shell-III and shell-IV respectively. From the Fig.1, it is clear that as frequency increases,

phase velocity, in general increases for composite spherical shell-I and shell-IV. For composite spherical shell-II and

shell-III, phase velocity, in general decreases. Fig. 2 depicts variation of attenuation with frequency for composite

spherical shell-I, shell-II, shell-III, and shell-IV. From the Fig. 2, it is seen that as frequency increases, attenuation,

in general increases for composite spherical shell-I, shell-II and shell-IV. In the case of composite spherical shell-III,

as frequency increases, attenuation, in general decreases. Also, attenuation of composite spherical shell-I are less

than that of shell-II and attenuation of composite spherical shell-III is less than that of shell-IV. If the shear viscosity

of fluid is ignored, the problem reduces to that of classical Biot’s theory. In this case phase velocity is computed as a

function of frequency and the results are depicted in Fig. 3. From this figure, it is observed that, as frequency

increases, phase velocity, in general increases for composite spherical shell-II and shell-III. For composite spherical

shell-I and shell-IV, phase velocity, in general decreases. Moreover, phase velocity of composite spherical shell-I

are greater than that of shell-II, and phase velocity of composite spherical shell-III are greater than that of shell-IV

for both extension theory and classical theory. In the particular case of thick walled poroelastic spherical shell, phase

velocity and attenuation are computed as a function of wavenumber, ratio of thickness to inner radius and the results

are depicted in Figs. 4 and 5. From the Fig. 4, it is observed that, as wavenumber increases, phase velocity, in

general increases for spherical shell-I and shell-III. From the Fig. 5, it is observed that, as ratio increases,

attenuation, in general decreases for spherical shell-I and increases for shell-III. From these Figs. 4 and 5, it is clear

that phase velocity and attenuation of spherical shell-I are less than that of shell-III.

Fig.1

Variation of phase velocity with frequency in the case of Biot’s

extension theory.

Parameters Berea Sandstone Shale Rock

Solid density s( ) . ( Kg / m ) 3 32 65 10 . ( Kg / m )31398 72

Solid frame shear modulus ( )0

. ( Kg / m sec ) 9 26 70 10 . ( Kg / m sec ) 8 27 59 10

Solid mineral shear modulus s( ) . Pa 102 30 10 . Pa 81 97 10

Solid frame bulk modulus ( K )0

. Pa 95 20 10 . Pa 937 91 10

Solid mineral bulk modulus ( K )0 Pa 935 10 . Pa 944 6 10

Porosity ( )0

.0 25 .0 15

Tortuosity ( s ) .1 33 .3 22

Permeability ( K ) . ( m ) 14 21 00 10 . ( m ) 18 21 973 10

Parameters Water Kerosene

Density f( ,Kg / m ) 3 . 31 00 10 .820 1

Shear viscosity f( ,Kg / m ) . 31 00 10 . 31 64 10

Bulk viscosity f( ,Kg / m ) 3 . 32 8 10 . 32 71 10

Bulk modulus f( K ,Pa) . 92 2 10 . 91 3 10



Fig.2

Variation of attenuation with frequency in the case of Biot’s

extension theory.

Fig.3

Variation of phase velocity with wavenumber in the case of

classical Biot’s theory.

Fig.4

Variation of phase velocity with wavenumber in the case of

thick walled poroelastic spherical shell.

Fig.5

Variation of attenuation with ratio in the case of thick walled

poroelastic spherical shell.

6 CONCLUSION

Employing the Biot’s extension theory, torsional vibrations of composite poroelastic spherical shell are investigated.

The frequency equation is investigated when the solids are saturated with viscous fluids. If the shear viscosity of

fluid is ignored, the problem reduces to that of classical Biot’s theory. Comparative study is made between the

extension theory and classical theory. From the numerical results, it is concluded that phase velocity of shell-I, shell-

III are higher than that of shell-II, shell-IV respectively for extension theory and classical theory. From this, it is

clear that shear viscosity of fluid is causing the discrepancy. Particular cases, namely, solid spherical shell and thick



walled hollow spherical shell are studied. Similar analysis can be made for any composite spherical shell which is

made of two different poroelastic solids if the pertinent values are available.

ACKNOWLEDGEMENTS

Authors acknowledge Department of Science and Technology (DST) for funding through Fund for Improvement of

S&T Infrastructure (FIST) program sanctioned to the Department of Mathematics, Kakatiya University, Warangal

with File No. SR/FST/MSI-101/2014. One of the authors Rajitha Gurijala acknowledges University Grants

Commission (UGC), Government of India, for its financial support through the Postdoctoral Fellowship for Women

(Grant number F.15-1/2015-17/PDFWM-2015-17-TEL-34525 (SA-II)).

APPENDIX A

mc ( )[ B ( kr )( Q ( D A D A ) Q ( D A D A )) B ( A N A N )

B ( kr )( Q ( D A D A ) Q ( D A D A )) B ( A N A N )],

11 2 1 2 1 10 10 20 20 2 10 20 20 10 1 10 10 20 20

2 2 1 10 20 20 10 2 10 10 20 20 2 10 20 20 10

mc ( )[ B ( kr )( Q ( N A N A ) Q ( N A A N )) B ( A D A D )

B ( kr )( Q ( A N A N ) Q ( N A N A )) B ( A D A D )],

12 2 1 2 1 2 1 1 2 2 2 1 2 1 1 1 1 2 2

2 2 1 1 2 2 1 2 1 1 2 2 2 1 2 2 1

c c , 13 16 0

c c , 23 26 0 mc ( )[ B Q ( kr )( G F G F ) B Q ( kr )( G F G F ) B ( G H G H )

B Q ( kr )( G F G F ) B Q ( kr )( G F G F ) B ( G H G H )],

14 2 3 10 2 10 10 20 20 3 20 2 10 20 20 10 3 10 10 20 20

4 10 2 10 20 20 10 4 20 2 10 10 20 20 4 10 20 20 10

mc ( )[ B Q ( kr )( G H G H ) B Q ( kr )( G H G H ) B ( E F E F )

B Q ( kr )( G H G H ) B Q ( kr )( G H G H ) B ( E F E F )],

15 2 3 10 2 2 2 1 1 3 20 2 1 2 2 1 3 1 1 2 2

4 20 2 2 2 1 1 4 10 2 1 2 2 1 4 1 2 2 1

i

i

c ( )[ B ( kr )Q ( A D A D ) B Q ( kr )( D A D A ) B ( A N A N )

B ( kr )Q ( D A D A ) B Q ( kr )( D A D A ) B ( A N A N )

( )( B Q ( kr )( D A D A ) B Q ( kr )( D A D A ) B ( A N A N )

B Q ( kr )( D A

21 2 5 2 1 1 1 2 2 5 2 2 1 2 2 1 5 1 1 2 2

6 2 1 1 2 2 1 6 2 2 1 1 2 2 6 1 2 2 1

2

5 1 2 1 2 2 1 5 2 2 1 1 2 2 5 1 2 2 1

6 1 2 1 1 D A ) B Q ( kr )( D A D A ) B ( A N A N ))], 2 2 6 2 2 1 2 2 1 6 1 1 2 2

i ic ( )[( )( B )( kr )( Q ( N A N A ) Q ( N A A N )) B ( A D A D )

B ( kr )( Q ( A N A N ) Q ( N A N A )) B ( A D A D )

B ( kr )( Q ( N A N A ) Q ( N A A N )) B ( A D A D )

B ( kr )( Q ( A N A N ) Q ( N

2

22 2 5 2 1 2 1 1 2 2 1 1 2 2 5 1 2 2 1

6 2 1 1 2 2 1 2 2 1 1 2 6 1 1 2 2

5 2 1 2 1 1 2 2 2 1 2 1 5 1 1 2 2

6 2 1 1 2 2 1 2 1A N A )) B ( A D A D )], 1 2 2 6 1 2 2 1

i

i

c ( )[ B Q ( kr )( G F G F ) B Q ( kr )( G F G F ) B ( G H G H )

B Q ( kr )( G F G F ) B Q ( kr )( G F G F ) B ( G H G H )

( )( B Q ( kr )( G F G F ) B Q ( kr )( G F G F ) B ( G H G H )

B Q ( kr

24 2 7 10 2 1 1 2 2 7 20 2 1 2 2 1 7 1 1 2 2

8 10 2 1 2 2 1 8 20 2 1 1 2 2 8 1 2 2 1

2

8 10 2 1 2 2 1 8 20 2 1 2 2 1 8 1 1 2 2

7 10 )( G F G F ) B Q ( kr )( G F G F ) B ( G H G H )], 2 1 2 2 1 7 20 2 1 1 2 2 7 1 2 2 1

i ic ( )[( ) B ( kr )( Q ( G H G H ) Q ( G H G H )) B ( E F E F )

B ( kr )( Q ( G H G H ) Q ( G H G H )) B ( E F E F )


B ( kr )( Q ( G H G H )

2

25 2 3 2 20 2 2 1 1 10 1 2 2 1 3 1 2 2 1

4 2 10 2 2 1 1 20 1 2 2 1 4 1 1 2 2

3 2 10 2 2 1 1 20 1 2 2 1 3 1 1 2 2

4 2 20 2 2 1 1 Q ( G H G H )) B ( E F E F )], 10 1 2 2 1 4 1 2 2 1

mc ( )[ B ( kr )( Q ( D A D A ) Q ( D A D A )) B ( A N A N )


31 2 1 1 1 1 1 2 2 2 1 2 2 1 1 1 1 2 2

2 1 1 1 2 2 1 2 1 1 2 2 2 1 2 2 1

mc ( )[ B ( kr )( Q ( N A N A ) Q ( N A A N )) B ( A D A D )


32 2 1 1 10 20 10 10 20 20 20 10 20 10 1 10 10 20 20

2 1 10 10 20 20 10 20 10 10 20 20 2 10 20 20 10

mc [ BY ( kr )( a d a d ) BY ( kr )( a d a d ) B ( a n a n )

B Y ( kr )( a d a d ) B Y ( kr )( a d a d ) B ( a n a n )],

33 1 1 11 1 10 10 20 20 1 22 1 10 20 20 10 1 10 10 20 20

2 11 1 10 20 20 10 2 22 1 10 10 20 20 2 10 20 20 10



mc ( )[ B Q ( kr )( G F G F ) B Q ( kr )( G F G F ) B ( G H G H )


34 2 3 10 1 10 10 20 20 3 20 1 10 20 20 10 3 10 10 20 20

4 10 1 10 20 20 10 4 20 1 10 10 20 20 4 10 20 20 10

mc ( )[ B Q ( kr )( G H G H ) B Q ( kr )( G H G H ) B ( E F E F )


35 2 3 10 1 20 20 10 10 3 20 1 10 20 20 10 3 10 10 20 20

4 20 1 20 20 10 10 4 10 1 10 20 20 10 4 10 20 20 10 mc [ B Y ( kr )( g f g f ) B Y ( kr )( g f g f ) B ( g h g h )

B Y ( kr )( g f g f ) B Y ( kr )( g f g f ) B ( g h g h )],

36 1 3 110 1 10 10 20 20 3 220 1 10 20 20 10 3 10 10 20 20

4 110 1 10 20 20 10 4 220 1 10 10 20 20 4 10 20 20 10

i

i

c ( )[ B ( kr )Q ( A D A D ) B Q ( kr )( D A D A ) B ( A N A N )

B ( kr )Q ( D A D A ) B Q ( kr )( D A D A ) B ( A N A N )

( )( B Q ( kr )( D A D A ) B Q ( kr )( D A D A

41 2 5 1 1 10 10 20 20 5 2 1 10 20 20 10 5 10 10 20 20

6 1 1 10 20 20 10 6 2 1 10 10 20 20 5 10 20 20 10

2

5 1 1 10 20 20 10 5 2 1 10 10 20 ) B ( A N A N )

B Q ( kr )( D A D A ) B Q ( kr )( D A D A ) B ( A N A N ))],

20 5 10 20 20 10

6 1 1 10 10 20 20 6 2 1 10 20 20 10 6 10 10 20 20

c42 have similar expressions as those of c41

with Q ,Q ,D ,D1 2 10 20 replaced by

Q ,Q ,N ,N10 20 10 20

i

i

c [ B Y ( kr )( a d a d ) B Y ( kr )( a d a d ) B ( a n a n )

B Y ( kr )( a d a d ) B Y ( kr )( a d a d ) B ( a n a n )

( )( B Y ( kr )( a d a d ) B Y ( kr )( a d a

43 1 5 11 1 10 10 20 20 5 22 1 10 20 20 10 5 10 10 20 20

6 11 1 10 20 20 10 6 22 1 10 10 20 20 6 10 20 20 10

1

5 11 1 10 20 20 10 5 22 1 10 10 d ) B ( a n a n )

B Y ( kr )( a d a d ) B Y ( kr )( a d a d ) B ( a n a n ))],

20 20 5 10 20 20 10

6 11 1 10 10 20 20 6 22 1 10 20 20 10 6 10 10 20 20

i

i



( )( B Q ( kr )( G F G F ) B Q ( kr )( G F G F ) B ( G H G H )

B Q ( kr

44 2 7 10 1 1 1 2 2 7 20 1 1 2 2 1 7 1 1 2 2

8 10 1 1 2 2 1 8 20 1 1 1 2 2 8 1 2 2 1

2

8 10 1 1 2 2 1 8 20 1 1 2 2 1 8 1 1 2 2

7 10 )( G F G F ) B Q ( kr )( G F G F ) B ( G H G H ))], 1 1 2 2 1 7 20 1 1 1 2 2 7 1 2 2 1

i

i

c ( )[ B Q ( kr )( G H G H ) B Q ( kr )( G H G H ) B ( E F E F )

B Q ( kr )( G H G H ) B Q ( kr )( G H G H ) B ( E F E F )

( )( B Q ( kr )( G H G H ) B Q ( kr )( G H

45 2 7 10 1 20 20 10 10 7 20 1 10 20 20 10 7 10 10 20 20

8 20 1 20 20 10 10 8 10 1 10 20 20 10 8 10 20 20 10

2

7 20 1 20 20 10 10 7 10 1 10 2 G H ) B ( E F E F )

B Q ( kr )( G H G H ) B Q ( kr )( G H G H ) B ( E F E F ))],

0 20 10 7 10 20 20 10

8 10 1 20 20 10 10 8 20 1 10 20 20 10 8 10 10 20 20

i

i



( )( B Q ( kr )( G F G F ) B Q ( kr )( G F

46 2 7 10 1 10 10 20 20 7 20 1 10 20 20 10 7 10 10 20 20

8 10 1 10 20 20 10 8 20 1 10 10 20 20 8 10 20 20 10

2

7 10 1 10 20 20 10 7 20 1 10 1 G F ) B ( G H G H )


0 20 20 7 10 20 20 10

8 10 1 10 10 20 10 8 20 1 10 20 20 10 8 10 10 20 20

c Z ( a n a n ) Z ( a n a n ), 51 1000 20 20 10 10 2000 10 20 20 10

c52 have similar expressions as those of c51 with n ,n10 20 replaced by d ,d10 20 ,

c Z ( A N A N ) Z ( A N A N ), 53 100 10 10 20 20 200 10 20 20 10

Z Z Z Z Z Z Z Zc ( g h g h ) ( g h g h ) ,

( Z ) ( Z ) ( Z ) ( Z )

3000 7000 8000 4000 3000 8000 7000 4000

54 20 20 10 10 10 20 20 102 2 2 2

3000 4000 3000 4000


with h ,h10 20 replaced by f , f10 20

,

Z Z Z Z Z Z Z Zc ( G H G H ) ( G H G H ) ,

( Z ) ( Z ) ( Z ) ( Z )

300 700 800 400 300 800 700 400

56 10 10 20 20 10 20 20 102 2 2 2

300 400 300 400

c61 have similar expressions as those of c51 with Z ,Z1000 2000 replaced by H ,H100 200 ,




with n ,n10 20 replaced by d ,d10 20

,

c H ( A N A N ) H ( A N A N ), 63 10 10 10 20 20 20 10 20 20 10

c Z ( g h g h ) Z ( g h g h ), 64 1000 10 10 20 20 2000 10 20 20 10



,

c Z (G H G H ) Z (G H G H ), 66 300 10 10 20 20 400 10 20 20 10 d d d d , 11 12 14 15 0

md ( )[ B ( kr )( Q ( D A D A ) Q ( D A D A )) B ( A N A N )


11 2 1 2 1 10 20 20 10 2 10 10 20 20 1 10 20 20 10

2 2 1 10 10 20 20 2 10 20 20 10 2 10 10 20 20

md ( )[( B )( kr )( Q ( N A N A ) Q ( N A A N )) B ( A D A D )


12 2 1 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1

2 2 1 1 2 2 1 2 2 1 1 2 2 1 1 2 2

d d , 13 16 0 d d , 23 26 0 md ( )[ B Q ( kr )( G F G F ) B Q ( kr )( G F G F ) B ( G H G H )


14 2 3 10 2 10 20 20 10 3 20 2 10 10 20 20 3 10 20 20 10

4 10 2 10 10 20 20 4 20 2 10 20 20 10 4 10 10 20 10

md ( )[ B Q ( kr )( G H G H ) B Q ( kr )( G H G H ) B ( E F E F )


15 2 3 20 2 2 2 1 1 3 10 2 1 2 2 1 3 1 2 2 1

4 10 2 2 2 1 1 4 20 2 1 2 2 1 4 1 1 2 2

i id ( )[( )( B ( kr )Q ( A D A D ) B Q ( kr )( D A D A ) B ( A N A N )

B ( kr )Q ( D A D A ) B Q ( kr )( D A D A ) B ( A N A N ))

( B Q ( kr )( D A D A ) B Q ( kr )( D A D A ) B ( A N A N )

B Q ( kr )( D

2

21 2 5 2 1 1 1 2 2 5 2 2 1 2 2 1 5 1 1 2 2

6 2 1 1 2 2 1 6 2 2 1 1 2 2 6 1 2 2 1

5 1 2 1 2 2 1 5 2 2 1 1 2 2 5 1 2 2 1

6 1 2 1A D A ) B Q ( kr )( D A D A ) B ( A N A N ))], 1 2 2 6 2 2 1 2 2 1 6 1 1 2 2

i

i

d ( )[( B )( kr )( Q ( N A N A ) Q ( N A A N )) B ( A D A D )

B ( kr )( Q ( A N A N ) Q ( N A N A )) B ( A D A D )

( )( B )( kr )( Q ( N A N A ) Q ( N A A N )) B ( A D A D )

B ( kr )( Q ( A N A N ) Q (

22 2 1 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1

2 2 1 1 2 2 1 2 2 1 1 2 2 1 1 2 2

2

1 2 1 2 1 1 2 2 2 1 2 1 1 1 1 2 2

2 2 1 1 2 2 1 2 N A N A )) B ( A D A D )], 1 1 2 2 2 1 2 2 1




B Q ( kr )( D

2

24 2 5 2 1 1 1 2 2 5 2 2 1 2 2 1 5 1 1 2 2

6 2 1 1 2 2 1 6 2 2 1 1 2 2 6 1 2 2 1

5 1 2 1 2 2 1 5 2 2 1 1 2 2 5 1 2 2 1


i id ( )[( ) B ( kr )( Q ( G H G H ) Q ( G H G H )) B ( E F E F )


( B ( kr )( Q ( G H G H ) Q ( G H G H )) B ( E F E F )

B ( kr )( Q ( G H G H )

2

25 2 3 2 10 2 2 1 1 20 1 2 2 1 3 1 1 2 2

4 2 20 2 2 1 1 10 1 2 2 1 4 1 2 2 1

3 2 20 2 2 1 1 10 1 2 2 1 3 1 2 2 1

4 2 10 2 2 1 1 Q ( G H G H )) B ( E F E F ))], 20 1 2 2 1 4 1 1 2 2

md [ B ( kr )( Q ( D A D A ) Q ( D A D A )) B ( A N A N )


31 2 1 1 1 1 2 2 1 2 1 1 2 2 1 1 2 2 1

2 1 1 1 1 2 2 2 1 2 2 1 2 1 1 2 2

md ( )[( B )( kr )( Q ( N A N A ) Q ( N A A N )) B ( A D A D )


32 2 1 1 10 20 10 10 20 20 10 10 20 20 1 10 20 20 10

2 1 10 10 20 20 10 20 20 10 10 20 2 10 10 20 20

md [ BY ( kr )( a d a d ) BY ( kr )( a d a d ) B ( a n a n )


33 1 1 11 1 10 20 20 10 1 22 1 10 10 20 20 1 10 20 20 10

2 11 1 10 10 20 20 2 22 1 10 20 20 10 2 10 10 20 20

md ( )[ B Q ( kr )( G F G F ) B Q ( kr )( G F G F ) B ( G H G H )


34 2 3 10 1 10 20 20 10 3 20 1 10 10 20 20 3 10 20 20 10

4 10 1 10 10 20 20 4 20 1 10 20 20 10 4 10 10 20 10



md ( )[ B Q ( kr )( G H G H ) B Q ( kr )( G H G H ) B ( E F E F )


35 2 3 20 1 20 20 10 10 3 10 1 10 20 20 10 3 10 20 20 10

4 10 1 20 20 10 10 4 20 1 10 20 20 10 4 10 10 20 20 md [ B Y ( kr )( g f g f ) B Y ( kr )( g f g f ) B ( g h g h )

B Y ( kr )( g f g f ) B Y ( kr )( g f g f ) B ( g h g h )],

36 1 3 110 1 10 20 20 10 3 220 1 10 10 20 20 3 10 20 20 10

4 110 1 10 10 20 20 4 220 1 10 20 20 10 4 10 10 20 20



( B Q ( kr )( D A D A ) B Q ( kr )( D A D

2

41 2 5 1 1 10 10 20 20 5 2 1 10 20 20 10 5 10 10 20 20

6 1 1 10 20 20 10 6 2 1 10 10 20 20 5 10 20 20 10

5 1 1 10 20 20 10 5 2 1 10 10 2 A ) B ( A N A N )

B Q ( kr )( D A D A ) B Q ( kr )( D A D A ) B ( A N A N ))],

0 20 5 10 20 20 10

6 1 1 10 10 20 20 6 2 1 10 20 20 10 6 10 10 20 20

d 42 have similar expressions as those of d 41

with Q ,Q ,D ,D1 2 10 20 replaced by Q ,Q ,N ,N ,10 20 10 20

i id [( )( B Y ( kr )( a d a d ) B Y ( kr )( a d a d ) B ( a n a n )

B Y ( kr )( a d a d ) B Y ( kr )( a d a d ) B ( a n a n ))

B Y ( kr )( a d a d ) B Y ( kr )( a d

1

43 1 5 11 1 10 10 20 20 5 22 1 10 20 20 10 5 10 10 20 20

6 11 1 10 20 20 10 6 22 1 10 10 20 20 6 10 20 20 10

5 11 1 10 20 20 10 5 22 1 10 10 a d ) B ( a n a n )


20 20 5 10 20 20 10

6 11 1 10 10 20 20 6 22 1 10 20 20 10 6 10 10 20 20




B Q ( kr )( D

2

44 2 5 1 1 1 1 2 2 5 2 1 1 2 2 1 5 1 1 2 2

6 1 1 1 2 2 1 6 2 1 1 1 2 2 6 1 2 2 1

5 1 1 1 2 2 1 5 2 1 1 1 2 2 5 1 2 2 1


i id ( )[( ) B Q ( kr )( G H G H ) B Q ( kr )( G H G H ) B ( E F E F )

B Q ( kr )( G H G H ) B Q ( kr )( G H G H ) B ( E F E F )

( B Q ( kr )( G H G H ) B Q ( kr )( G H

2

45 2 7 10 1 20 20 10 10 7 20 1 10 20 20 10 7 10 10 20 20

8 20 1 20 20 10 10 8 10 1 10 20 20 10 8 10 20 20 10

7 20 1 20 20 10 10 7 10 1 10 2 G H ) B ( E F E F )

B Q ( kr )( G H G H ) B Q ( kr )( G H G H ) B ( E F E F ))],

0 20 10 7 10 20 20 10

8 10 1 20 20 10 10 8 20 1 10 20 20 10 8 10 10 20 20

i id [( )( B Y ( kr )( g f g f ) B Y ( kr )( g f g f ) B ( g h g h )

B Y ( kr )( g f g f ) B Y ( kr )( g f g f ) B ( g h g h ))

( B Y ( kr )( g f g f ) B Y ( kr )(

1

46 1 7 110 1 10 10 20 20 7 220 1 10 20 20 10 7 10 10 20 20

8 110 1 10 20 20 10 8 220 1 10 10 20 20 8 10 20 20 10

7 110 1 10 20 20 10 7 220 1 g f g f ) B ( g h g h )

B Y ( kr )( g f g f ) B Y ( kr )( g f g f ) B ( g h g h ))],

10 10 20 12 7 10 20 20 10

8 110 1 10 10 20 20 8 220 1 10 20 20 10 8 10 10 20 20

d ( Z )( a n a n ) Z ( a n a n ), 51 1000 10 20 20 10 2000 10 10 20 20


with n ,n10 20 replaced by d ,d10 20

,

d Z ( A N A N ) Z ( A N A N ), 53 100 10 20 20 10 200 10 10 20 20

Z Z Z Z Z Z Z Zd ( g h g h ) ( g h g h ) ,

( Z ) ( Z ) ( Z ) ( Z )

3000 7000 8000 4000 3000 8000 7000 4000

54 10 20 20 10 20 20 10 102 2 2 2

3000 4000 3000 4000



,

Z Z Z Z Z Z Z Zd ( G H G H ) ( G H G H ) ,

( Z ) ( Z ) ( Z ) ( Z )

300 700 800 400 300 800 700 400

56 10 20 20 10 10 10 20 202 2 2 2

300 400 300 400


with Z ,Z1000 2000 replaced by H ,H100 200 ,


with n ,n10 20 replaced by



d ,d10 20, d H ( A N A N ) H ( A N A N ), 63 10 10 20 20 10 20 10 10 20 20

d 64 have similar expressions as those of c64

with Z ,Z1000 2000 replaced by Z , Z2000 1000

,



,



.

In all the above equations, the notation expressions are as follows:

c c f

c c f

f

mB , B , B m , B ,

m

2 21 12 2 1 1 1 0

1 1 2 3 1 1 4 1

1 1 1

c f c s

i i

B d d ,

2

2 2

5 1 1 1 1 1

1 1 1

c s f c

i i

B d d ,

2

2 2

6 1 1 1 1 1

1 1 1

c f f s

i i f

B m d d ,m

2

2 1 0

7 1 1 1 1 1

1 1 1 1

c f s f

i f i

B m d d ,m

2

2 1 0

8 1 1 1 1 1

1 1 1 1

B ,B ,B ,B ,B ,B ,B ,B10 20 30 40 50 60 70 80 are similar expressions as B ,B ,B ,B ,B ,B ,B ,B1 2 3 4 5 6 7 8 with left subscript

1 replaced by 2,

c f f

c f f f f

i i i

d mL , L , L , L ( d m ) d ,

22 2

2 2 2 2

1 2 2 3 4 2 2 2 2

2 2 2

1

c f f f f

c f f

i i i

d m dL , L , L ( d m ),

2

2 2 2 2 2

10 2 20 30 2 2

2 2 2

12

f f

i

dL ( ) ,

2 2

40

2

/ VW V V cos tan ,

V

2 2 1 2 1 2

1 1 2

1

1

2 / V

W V V sin tan ,V

2 2 1 2 1 2

2 1 2

1

1

2

W , W11 22are similar expressions as W , W1 2 with V ,V1 2

replaced by V ,V ,11 22

c f f f f

c f f f f

i i i

d m dV (V ), V , V V , V ( d m ) d ,

22 2

2 2 22 2 2 2 2

1 2 2 11 22 2 2 2 2

2 2 2

1

W L W LQ ,

L L

1 10 2 20

1 2 2

10 20

W L W LQ ,

L L

2 10 1 20

2 2 2

10 20

W L W LQ ,

L L

11 30 22 40

10 2 2

30 40

W L W LQ ,

L L

22 30 11 40

20 2 2

30 40

R ,R ,R ,R1 2 3 4 are similar expressions as L ,L ,L ,L1 2 3 4

with left subscript 2 replaced by 1,

R ,R ,R ,R10 20 30 40 are similar expressions as L ,L ,L ,L10 20 30 40


/ oc o o cos tan ,

o

2 2 1 2 1 2

1 1 2

1

1

2 / o

c o o sin tan ,o

2 2 1 2 1 2

2 1 2

1

1

2

c , c11 22 are similar expressions as c , c1 2 with o , o1 2

replaced by o , o ,11 22

o , o ,1 2o , o ,11 22 are similar expressions as V ,V ,1 2

V ,V ,11 22 with left subscript 2 replaced by 1,

c R c RY ,

R R

1 10 2 20

11 2 2

10 20

c R c RY ,

R R

2 10 1 20

22 2 2

10 20

W R W RY ,

R R

110 30 220 40

110 2 2

30 40

W R W RY ,

R R

220 30 110 40

220 2 2

30 40



W ,W110 220 are similar expressions as W ,W11 22


Ya cos tan ,

YY Y kr

1 22

102 2

1111 22 1

1 1

2

Ya sin tan ,

YY Y kr

1 22

202 2

1111 22 1

1 1

2

n cos(Y kr ) cosh(Y kr ) sin(Y kr ) cosh(Y kr ), 10 11 1 22 1 11 1 22 1

n cos(Y kr ) sinh(Y kr ) sin(Y kr ) sinh(Y kr ), 20 11 1 22 1 11 1 22 1

a ,a ,n ,n1 2 1 2 are similar expressions as a ,a ,n ,n10 20 10 20

with r1replaced by r ,2

A ,A10 20, N ,N10 20

are similar expressions as a ,a10 20, n ,n10 20

with Y ,Y11 22replaced by Q ,Q ,1 2

A ,A ,N ,N1 2 1 2 are similar expressions as A ,A ,N ,N10 20 10 20


g , g ,h ,h10 20 10 20 are similar expressions as a ,a ,n ,n10 20 10 20

with Y ,Y11 22replaced by Y ,Y ,110 220

g , g ,h ,h1 2 1 2 are similar expressions as g , g ,h ,h10 20 10 20


G ,G ,H ,H10 20 10 20 are similar expressions as g , g ,h ,h10 20 10 20


G ,G ,H ,H1 2 1 2 are similar expressions as G ,G ,H ,H10 20 10 20


d sin(Y kr ) cosh(Y kr ) cos(Y kr ) cosh(Y kr ), 10 11 1 22 1 11 1 22 1

d cos(Y kr ) sinh(Y kr ) sin(Y kr ) sinh(Y kr ), 20 11 1 22 1 11 1 22 1

d ,d ,1 2 are similar expressions as d ,d10 20


D ,D ,D ,D10 20 1 2 are similar expressions as d ,d ,d ,d10 20 1 2


f , f10 20 are similar expressions as d ,d1 2

with Y ,Y11 22replaced by Y ,Y ,110 220

f , f1 2 are similar expressions as f , f10 20


F ,F ,F ,F10 20 1 2 are similar expressions as f , f ,10 20 f , f1 2

with Y ,Y110 220 replaced by Q ,Q ,10 20

X cos(Y kr ) cosh(Y kr ) sin(Y kr ) cosh(Y kr ), 10 110 1 220 1 110 1 220 1

X cos(Y kr ) sinh(Y kr ) sin(Y kr ) sinh(Y kr ), 20 110 1 220 1 110 1 220 1

X ,X1 2 are similar expressions as X ,X10 20


/ lZ l l cos tan ,

l

2 2 1 2 1 2

100 1 2

1

1

2 / l

Z l l sin tan ,l

2 2 1 2 1 2

200 1 2

1

1

2

( L B )( L L ) ( L B )( L L )l ,

( L L ) ( L L )

1 70 1 3 2 80 2 4

1 2 2

1 3 2 4

( L B )( L L ) ( L B )( L L )l ,

( L L ) ( L L )

2 80 1 3 1 70 2 4

2 2 2

1 3 2 4

/ lZ l l cos tan ,

l

2 2 1 2 1 4

1000 3 4

3

1

2 / l

Z l l sin tan ,l

2 2 1 2 1 4

2000 3 4

3

1

2

( R B )( R R ) ( R B )( R R )l ,

( R R ) ( R R )

1 7 1 3 2 8 2 4

3 2 2

1 3 2 4

( R B )( R R ) ( R B )( R R )l ,

( R R ) ( R R )

2 8 1 3 1 7 2 4

4 2 2

1 3 2 4

/ pZ p p cos tan ,

p

2 2 1 2 1 2

300 1 2

1

1

2 / p

Z p p sin tan ,p

2 2 1 2 1 2

400 1 2

1

1

2

( B L )( L L ) ( B L )( L L )p ,

( L L ) ( L L )

10 3 1 3 20 4 2 4

1 2 2

1 3 2 4

( B L )( L L ) ( B L )( L L )P ,

( L L ) ( L L )

20 4 1 3 10 3 4 2

2 2 2

1 3 2 4

/ lZ l l cos tan ,

l

2 2 1 2 1 200

3000 100 200

100

1

2 / l

Z l l sin tan ,l

2 2 1 2 1 200

4000 100 200

100

1

2



m ( R R ) m ( R R )l ,

( R R ) ( R R )

100 3 1 200 4 2

100 2 2

1 3 2 4

m ( R R ) m ( R R )l ,

( R R ) ( R R )

100 2 4 200 3 1

200 2 2

1 3 2 4

f f

c

i

dm ,

2

21 1

100 1

1

c

f f f f

i

m d m d ,

231

200 1 1 1 12

11

1

B m B m

Z ,m m

3 100 4 200

7000 2 2

100 200

B m B mZ ,

m m

4 100 3 200

8000 2 2

100 200

B m B mZ ,

m m

30 10 40 20

700 2 2

10 20

B m B mZ ,

m m

40 10 30 20

800 2 2

10 20

m ,m10 20 are similar expressions as m ,m100 200


Z Z Z ZH ,

( Z ) ( Z )

5000 1000 6000 2000

100 2 2

1000 2000

Z Z Z ZH ,

( Z ) ( Z )

6000 1000 5000 2000

200 2 2

1000 2000

Z Z Z ZH ,

( Z ) ( Z )

500 100 600 200

10 2 2

100 200

Z Z Z ZH ,

( Z ) ( Z )

600 100 500 200

20 2 2

100 200

a BZ ,

B B

500 4

5000 2 2

3 4

a BZ ,

B B

500 3

6000 2 2

3 4

c c f f

i

a d m ,

2 2

500 1 1 1 1 1

1 1

a BZ ,

B B

50 40

500 2 2

30 40

a BZ ,

B B

50 30

600 2 2

30 40

a50are similar expressions as a500

with left subscript 1 replaced by 2.

REFERENCES

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44 (3): 749-751.

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Journal of Engineering Mathematics 2014: 416406.

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framework of Biot’s extension theory, Multidiscipline Modeling in Materials and Structures 14(5): 970-983.

[14] Milton A., Irene A.S., 1964, Handbook of Mathematical Functions, Dover Publications, National Bureau of Standard

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study of torsional vibrations of composite poroelastic

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